What is (-3+sqrt(3)i)^7 ?

The solution to (-3+sqrt(3)i)^7 is 5184-1728sqrt(3)i

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simplify (-3+sqrt(3)i)^7

simplify

(- 3 + 3 i)^{7}

5184 - 17283 i

Solution steps

(- 3 + 3 i)^{7}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = - 3, b = 3 i

= i = 0 \sum 7 (i 7) (- 3)^{(7 - i)} (3 i)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} (- 3)^{7} (3 i)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (- 3)^{6} (3 i)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (- 3)^{5} (3 i)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (- 3)^{4} (3 i)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (- 3)^{3} (3 i)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (- 3)^{2} (3 i)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (- 3)^{1} (3 i)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (- 3)^{0} (3 i)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (- 3)^{7} (3 i)^{0} : - 2187

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (- 3)^{6} (3 i)^{1} : 51033 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (- 3)^{5} (3 i)^{2} : - 15309 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (- 3)^{4} (3 i)^{3} : 85053 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} (- 3)^{3} (3 i)^{4} : - 8505 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} (- 3)^{2} (3 i)^{5} : 17013 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} (- 3)^{1} (3 i)^{6} : - 567 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (- 3)^{0} (3 i)^{7} : 273 i^{7}

= - 2187 + 51033 i - 15309 i^{2} + 85053 i^{3} - 8505 i^{4} + 17013 i^{5} - 567 i^{6} + 273 i^{7}

Simplify

- 2187 + 51033 i - 15309 i^{2} + 85053 i^{3} - 8505 i^{4} + 17013 i^{5} - 567 i^{6} + 273 i^{7} : - 17283 i + 5184

= - 17283 i + 5184

Rewrite in standard complex form:

5184 - 17283 i

= 5184 - 17283 i

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